A physical quantity X is related to three observables a, b, c as X = √a b²/c². The errors of measurement in a, b and c are 2%, 1% and 3% respectively. What is the percentage error in the quantity X?

Correct Option: 9%

To find the percentage error in the physical quantity $X$, we start with the given relationship between $X$ and the observables $a$, $b$, and $c$:

$X=\frac{\sqrt{a}{b}^2}{c^2}=a^{1/2}{b}^2{c}^{-2}$

When calculating errors, the relative error in a product or quotient is the sum of the relative errors of each individual quantity multiplied by their respective powers. Note that errors are always added to find the maximum possible error, so the negative sign in the exponent of $c$ becomes positive when summing errors.

Therefore, the fractional error in $X$ is given by:

$\frac{\Delta X}{X}=\frac{1}{2}\left(\frac{\Delta a}{a}\right)+2\left(\frac{\Delta b}{b}\right)+2\left(\frac{\Delta c}{c}\right)$

To express this in terms of percentage error, we multiply the entire equation by 100:

$\left(\frac{\Delta X}{X}\times 100\right)\%=\left[\frac{1}{2}\left(\frac{\Delta a}{a}\times 100\right)+2\left(\frac{\Delta b}{b}\times 100\right)+2\left(\frac{\Delta c}{c}\times 100\right)\right]\%$

We are given the percentage errors for each of the observables:
- Percentage error in $a$ = $\left(\frac{\Delta a}{a}\times 100\right)\%=2\%$
- Percentage error in $b$ = $\left(\frac{\Delta b}{b}\times 100\right)\%=1\%$
- Percentage error in $c$ = $\left(\frac{\Delta c}{c}\times 100\right)\%=3\%$

Substituting these values into our expression:

$\text{Percentage error in}X=\left[\frac{1}{2}\left(2\right)+2(1)+2(3)\right]\%$

$\text{Percentage error in}X=\left[1+2+6\right]\%=9\%$

Thus, the percentage error in the quantity $X$ is 9%.